Pseudorandomness and Average-Case
Complexity via Uniform Reductions

Luca Trevisan and
Salil Vadhan

Abstract

Impagliazzo and Wigderson (36th FOCS, 1998) gave the first
construction of pseudorandom generators from a *uniform* complexity assumption
on EXP (namely EXP\neq BPP). Unlike results in the nonuniform setting, their
result does not provide a continuous trade-off between worst-case hardness and
pseudorandomness, nor does it explicitly establish an average-case hardness
result.

In this paper:

We obtain an optimal
worst-case to average-case connection for EXP: if EXP is not a
subset of BPTIME(t(n)), EXP has problems that cannot be solved on a
fraction 1/2 +1/t'(n) of the inputs by BPTIME(t'(n)) algorithms, for
t'=t^{\Omega(1)}.

We exhibit a PSPACE-complete
self-correctible and downward self-reducible problem. This slightly
simplifies and strengthens the proof of Impaglaizzo and Wigderson, which
used a a #P-complete problem with these properties.

We argue that the results of
Impagliazzo and Wigderson, and the ones in this paper, cannot be proved
via "black-box" uniform reductions.